Differential and numerical models of hysteretic systems with stochastic and deterministic inputs

Many deterministic models with hysteresis have been developed in the areas of economics, finance, terrestrial hydrology and biology. These models lack any stochastic element which can often have a strong effect in these areas. In this work stochastically driven closed loop systems with hysteresis type memory are studied. This type of system is presented as a possible stochastic counterpart to deterministic models in the areas of economics, finance, terrestrial hydrology and biology. Some price dynamics models are presented as a motivation for the development of this type of model. Numerical schemes for solving this class of stochastic differential equation are developed in order to examine the prototype models presented. As a means of further testing the developed numerical schemes, numerical examination is made of the behaviour near equilibrium of coupled ordinary differential equations where the time derivative of the Preisach operator is included in one of the equations. A model of two phenotype bacteria is also presented. This model is examined to explore memory effects and related hysteresis effects in the area of biology. The memory effects found in this model are similar to that found in the non-ideal relay. This non-ideal relay type behaviour is used to model a colony of bacteria with multiple switching thresholds. This model contains a Preisach type memory with a variable Preisach weight function. Shown numerically for this multi-threshold model is a pattern formation for the distribution of the phenotypes among the available thresholds.

Estimation of atmospheric nutrient inputs to the Atlantic Ocean from 50°N to 50°S based on large-scale field sampling: Iron and other dust-associated elements

Atmospheric inputs of mineral dust supply iron and other trace metals to the remote ocean and can influence the marine carbon cycle due to iron's role as a potentially limiting micronutrient. Dust generation, transport, and deposition are highly heterogeneous, and there are very few remote marine locations where dust concentrations and chemistry (e.g., iron solubility) are routinely monitored. Here we use aerosol and rainwater samples collected during 10 large-scale research cruises to estimate the atmospheric input of iron, aluminum, and manganese to four broad regions of the Atlantic Ocean over two 3 month periods for the years 2001–2005. We estimate total inputs of these metals to our study regions to be 4.2, 17, and 0.27 Gmol in April–June and 4.9, 14, and 0.19 Gmol in September–November, respectively. Inputs were highest in regions of high rainfall (the intertropical convergence zone and South Atlantic storm track), and rainfall contributed higher proportions of total input to wetter regions. By combining input estimates for total and soluble metals for these time periods, we calculated overall percentage solubilities for each metal that account for the contributions from both wet and dry depositions and the relative contributions from different aerosol types. Calculated solubilities were in the range 2.4%–9.1% for iron, 6.1%–15% for aluminum, and 54%–73% for manganese. We discuss sources of uncertainty in our estimates and compare our results to some recent estimates of atmospheric iron input to the Atlantic.

A framework for managing uncertain inputs: An axiomization of rewarding

The success postulate in belief revision ensures that new evidence (input) is always trusted. However, admitting uncertain input has been questioned by many researchers. Darwiche and Pearl argued that strengths of evidence should be introduced to determine the outcome of belief change, and provided a preliminary definition towards this thought. In this paper, we start with Darwiche and Pearl’s idea aiming to develop a framework that can capture the influence of the strengths of inputs with some rational assumptions. To achieve this, we first define epistemic states to represent beliefs attached with strength, and then present a set of postulates to describe the change process on epistemic states that is determined by the strengths of input and establish representation theorems to characterize these postulates. As a result, we obtain a unique rewarding operator which is proved to be a merging operator that is in line with many other works. We also investigate existing postulates on belief merging and compare them with our postulates. In addition, we show that from an epistemic state, a corresponding ordinal conditional function by Spohn can be derived and the result of combining two epistemic states is thus reduced to the result of combining two corresponding ordinal conditional functions proposed by Laverny and Lang. Furthermore, when reduced to the belief revision situation, we prove that our results induce all the Darwiche and Pearl’s postulates as well as the Recalcitrance postulate and the Independence postulate.